A ratio-cum-product estimator of population mean in stratified random sampling using two auxiliary variables

STATISTICA, anno LXXII, n. 3, 2012 A RATIO-CUM-PRODUCT ESTIMATOR OF POPULATION MEAN IN STRATIFIED RANDOM SAMPLING USING TWO AUXILIARY VARIABLES R. Tailor, S. Chouhan, R. Tailor, N. Garg 1. INTRODUCTION A country or state frequently requires estimates of agricultural production to assess status of grain and to make future policies regarding export and import of grains according to need. It requires the estimates of total production, average production and per hectare production of any crop which corresponds to the problem of estimation of population total, population mean and ratio of two population means respectively. This paper discusses the problem of estimation of finite population mean using information on two auxiliary variates. Auxiliary information is often used by researchers in order to improve the efficiencies of estimators. Cochran (1940) used auxiliary information at estimation stage and envisaged ratio method of estimation that provides ratio estimator. Ratio estimator has higher efficiency when study variate and auxiliary variates are positively correlated. Robson (1957) developed product method of estimation that provides product estimator. When study variate and auxiliary variate are negatively correlated, product estimator gives higher efficiency in comparison to simple mean estimator provided correlation coefficient between study variate and auxiliary variate is greater than half of the ratio of coefficient of variation of auxiliary variate and coefficient of variation of study variate. In both, ratio and product methods of estimation, population mean of the auxiliary variate is assumed to be known. Singh (1967) utilized information on two auxiliary variates, one is positively correlated and another is negatively correlated with the study variate and suggested ratio-cum-product estimator of population mean in simple random sampling. Later many authors proposed various ratio and product type estimators in simple random sampling, for instance see Sisodia and Dwivedi (1981), Pandey and Dubey (1988), Upadhyaya and Singh (1999), Singh and Tailor (2003), Singh et al. (2004), Singh and Tailor (2005), Kadilar and Cingi (2006), Singh et al. (2009), Singh et al. (2011), etc.. Hansen et al. (1946) defined combined ratio estimator using auxiliary information in stratified random sampling. Many authors including Kadilar and Cingi (2003, 2005, 2006) and Singh et al. (2008) worked out ratio type estimators in stratified random sampling. 288 R. Tailor, S. Chouhan, R. Tailor, N. Garg Simple random sampling technique has some shortcomings like less representative of different sections of the population, administrative inconvenience and less efficiency in case of heterogeneous population. Literature reveals that ratiocum-product estimator performs better than ratio and product type estimators in simple random sampling under certain conditions. This motivates authors to work out Singh (1967) ratio-cum-product estimator in stratified random sampling and study its properties. Consider a finite population U  {U1 , U 2 ,..., U N } of size N and it is divided into L strata of size N h ( h  1, 2,..., L ). Let Y be the study variate and X and Z be two auxiliary variates taking values yhi , x hi and z hi ( h  1, 2,..., L ; i  1, 2,..., N h ) on i th unit of the h th stratum. A sample of size nh L is drawn from each stratum which constitutes a sample of size n   nh and we h 1 define: Yh  1 Nh yhi : h th stratum mean for the study variate Y ,  N h i 1 Xh  1 Nh  x hi : h th stratum mean for the auxiliary variate X , N h i 1 Zh  1 Nh  z hi : h th stratum mean for the auxiliary variate Z , N h i 1 Y L 1 L Nh 1 L y hi   N h Yh  Wh Yh : population mean of the study vari N h 1 i 1 N h 1 h 1 ate Y , X 1 L Nh 1 L x hi  Wh X h : population mean of the auxiliary variate X ,  N h 1 i 1 N h 1 Z 1 L Nh 1 L z   hi N Wh Zh : population mean of the auxiliary variate Z , N h 1 i 1 h 1 yh  1 nh yhi : sample mean of the study variate Y for h th stratum,  nh i 1 xh  1 nh x hi : sample mean of the auxiliary variate X for h th stratum,  nh i 1 A ratio-cum-product estimator of population mean etc. zh  289 1 nh z hi : sample mean of the auxiliary variate Z for h th stratum, nh i 1 Wh  Nh : stratum weight of h th stratum. N Usual unbiased estimators of population means Y , X and Z in stratified random sampling are defined respectively as L y st   Wh y h , (1) h 1 L x st   Wh x h , (2) h 1 L z st   Wh z h . (3) h 1 In the line of Cochran (1940) ratio estimator, Hansen et al. (1946) utilized known value of population mean X of auxiliary variate X and defined combined ratio estimator for population mean Y as X  YˆRC  y st  .  x st  (4) Here it is assumed that the study variate Y and the auxiliary variate X are positively correlated. When the study variate Y and the auxiliary variate Z are negatively correlated, assuming that the population mean Z of auxiliary variate Z is known, combined product estimator is defined as z  YˆPC  y st  st  . Z  (5) The bias and mean squared error of YˆRC and YˆPC , up to the first degree of approximation, are obtained as B( YˆRC )  1 L 2 Wh  h ( R1Sxh2  S yxh ) , X h 1 (6) 290 R. Tailor, S. Chouhan, R. Tailor, N. Garg B( YˆPC )  1 L 2 Wh  h S yzh , Z h 1 L 2 MSE( YˆRC )   Wh2 h ( S 2yh  R12 S xh  2R1S yxh ) , (7) (8) h 1 L MSE( YˆPC )   Wh2 h ( S 2yh  R 22 Szh2  2R 2 S yzh ) (9) h 1 where S 2yh  Szh2  1 Nh 1 Nh 2 ( ) z  Z S  ,  hi h  ( yhi  Yh )( x hi  X h ) , yxh N h  1 i 1 N h  1 i 1 S yzh  R1  1 Nh 1 Nh 2 ( y hi  Yh )2 , Sxh    ( x hi  X h )2 , N h  1 i 1 N h  1 i 1 1 Nh 1 Nh ( )( ) y  Y z  Z S  ,  hi h hi h xzh N  1  ( x hi  X h )(z hi  Zh ) , N h  1 i 1 i 1 h  1 1  Y Y , R2  and  h    . X Z  nh N h  2. PROPOSED ESTIMATOR Assuming that the population means of auxiliary variates X and Z are known, Singh (1967) proposed a ratio-cum-product estimator for population mean Y as  X  z  YˆRP  y      x  Z  (10) 1 n 1 n y and x   i  x i are unbiased estimates of population means n i 1 n i 1 Y and X in simple random sampling without replacement. where y  We propose Singh (1967) ratio-cum- product estimator YˆRP in stratified random sampling as 291 A ratio-cum-product estimator of population mean etc.  L  L  W X Wh z h    h h   L  X   z st  ST h 1  h 1 . YˆRP  y st       Wh Yh  L   L   x st   Z  h 1   Wh x h   Wh Zh   h 1  h 1  (11) To compare the efficiency of the proposed estimator in comparison to other estimators, bias and mean squared error of the proposed estimator are obtained. To obtain the bias and mean squared error expressions of the proposed estimator YˆRPST , we write y h  Yh (1  e 0 h ), x h  X h (1  e1 h ) and z h  Z h (1  e 2 h ) such that E( e 0h )  E( e1h )  E( e 2h )  (...truncated)